Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/166

into two branches, so that when one branch is opened to let the magnet pass the current continues to flow through the other. Faraday used for this purpose a circular trough of mercury, as shewn in Fig. 23, Art. 491. The current enters the trough through the wire $AB$, it is divided at $B$, and after flowing through the arcs $BQP$ and $BRP$ it unites at $P$, and leaves the trough through the wire $PO$, the cup of mercury $O$, and a vertical wire beneath $O$, down which the current flows.

The magnet (not shewn in the figure) is mounted so as to be capable of revolving about a vertical axis through $O$, and the wire $OP$ revolves with it. The body of the magnet passes through the aperture of the trough, one pole, say the north pole, being beneath the plane of the trough, and the other above it. As the magnet and the wire $OP$ revolve about the vertical axis, the current is gradually transferred from the branch of the trough which lies in front of the magnet to that which lies behind it, so that in every complete revolution the magnet passes from one side of the current to the other. The north pole of the magnet revolves about the descending current in the direction N.E.S.W. and if $\omega$, $\omega ^{\prime }$ are the solid angles (irrespective of sign) subtended by the circular trough at the two poles, the work done by the electromagnetic force in a complete revolution is

where $m$ is the strength of either pole, and $i$ the strength of the current.

487.] Let us now endeavour to form a notion of the state of the magnetic field near a linear electric circuit.

Let the value of $\omega$, the solid angle subtended by the circuit, be found for every point of space, and let the surfaces for which $\omega$ is constant be described. These surfaces will be the equipotential surfaces. Each of these surfaces will be bounded by the circuit, and any two surfaces, $\omega _{1}$ and $\omega _{2}$, will meet in the circuit at an angle ${\frac {1}{2}}(\omega _{1}-\omega _{2})$.

Figure XVIII, at the end of this volume, represents a section of the equipotential surfaces due to a circular current. The small circle represents a section of the conducting wire, and the horizontal line at the bottom of the figure is the perpendicular to the plane of the circular current through its centre. The equipotential surfaces, 24 of which are drawn corresponding to a series of values of $\omega$ differing by ${\frac {\pi }{6}}$, are surfaces of revolution, having this line for